Properties

Label 147.4.a.b.1.1
Level $147$
Weight $4$
Character 147.1
Self dual yes
Analytic conductor $8.673$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 147 = 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 147.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(8.67328077084\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 21)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 147.1

$q$-expansion

\(f(q)\) \(=\) \(q-3.00000 q^{2} +3.00000 q^{3} +1.00000 q^{4} -3.00000 q^{5} -9.00000 q^{6} +21.0000 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q-3.00000 q^{2} +3.00000 q^{3} +1.00000 q^{4} -3.00000 q^{5} -9.00000 q^{6} +21.0000 q^{8} +9.00000 q^{9} +9.00000 q^{10} -15.0000 q^{11} +3.00000 q^{12} -64.0000 q^{13} -9.00000 q^{15} -71.0000 q^{16} +84.0000 q^{17} -27.0000 q^{18} -16.0000 q^{19} -3.00000 q^{20} +45.0000 q^{22} -84.0000 q^{23} +63.0000 q^{24} -116.000 q^{25} +192.000 q^{26} +27.0000 q^{27} -297.000 q^{29} +27.0000 q^{30} -253.000 q^{31} +45.0000 q^{32} -45.0000 q^{33} -252.000 q^{34} +9.00000 q^{36} -316.000 q^{37} +48.0000 q^{38} -192.000 q^{39} -63.0000 q^{40} +360.000 q^{41} +26.0000 q^{43} -15.0000 q^{44} -27.0000 q^{45} +252.000 q^{46} -30.0000 q^{47} -213.000 q^{48} +348.000 q^{50} +252.000 q^{51} -64.0000 q^{52} +363.000 q^{53} -81.0000 q^{54} +45.0000 q^{55} -48.0000 q^{57} +891.000 q^{58} -15.0000 q^{59} -9.00000 q^{60} -118.000 q^{61} +759.000 q^{62} +433.000 q^{64} +192.000 q^{65} +135.000 q^{66} -370.000 q^{67} +84.0000 q^{68} -252.000 q^{69} -342.000 q^{71} +189.000 q^{72} +362.000 q^{73} +948.000 q^{74} -348.000 q^{75} -16.0000 q^{76} +576.000 q^{78} +467.000 q^{79} +213.000 q^{80} +81.0000 q^{81} -1080.00 q^{82} +477.000 q^{83} -252.000 q^{85} -78.0000 q^{86} -891.000 q^{87} -315.000 q^{88} +906.000 q^{89} +81.0000 q^{90} -84.0000 q^{92} -759.000 q^{93} +90.0000 q^{94} +48.0000 q^{95} +135.000 q^{96} +503.000 q^{97} -135.000 q^{99} +O(q^{100})\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −3.00000 −1.06066 −0.530330 0.847791i \(-0.677932\pi\)
−0.530330 + 0.847791i \(0.677932\pi\)
\(3\) 3.00000 0.577350
\(4\) 1.00000 0.125000
\(5\) −3.00000 −0.268328 −0.134164 0.990959i \(-0.542835\pi\)
−0.134164 + 0.990959i \(0.542835\pi\)
\(6\) −9.00000 −0.612372
\(7\) 0 0
\(8\) 21.0000 0.928078
\(9\) 9.00000 0.333333
\(10\) 9.00000 0.284605
\(11\) −15.0000 −0.411152 −0.205576 0.978641i \(-0.565907\pi\)
−0.205576 + 0.978641i \(0.565907\pi\)
\(12\) 3.00000 0.0721688
\(13\) −64.0000 −1.36542 −0.682708 0.730691i \(-0.739198\pi\)
−0.682708 + 0.730691i \(0.739198\pi\)
\(14\) 0 0
\(15\) −9.00000 −0.154919
\(16\) −71.0000 −1.10938
\(17\) 84.0000 1.19841 0.599206 0.800595i \(-0.295483\pi\)
0.599206 + 0.800595i \(0.295483\pi\)
\(18\) −27.0000 −0.353553
\(19\) −16.0000 −0.193192 −0.0965961 0.995324i \(-0.530796\pi\)
−0.0965961 + 0.995324i \(0.530796\pi\)
\(20\) −3.00000 −0.0335410
\(21\) 0 0
\(22\) 45.0000 0.436092
\(23\) −84.0000 −0.761531 −0.380765 0.924672i \(-0.624339\pi\)
−0.380765 + 0.924672i \(0.624339\pi\)
\(24\) 63.0000 0.535826
\(25\) −116.000 −0.928000
\(26\) 192.000 1.44824
\(27\) 27.0000 0.192450
\(28\) 0 0
\(29\) −297.000 −1.90178 −0.950888 0.309535i \(-0.899827\pi\)
−0.950888 + 0.309535i \(0.899827\pi\)
\(30\) 27.0000 0.164317
\(31\) −253.000 −1.46581 −0.732906 0.680330i \(-0.761836\pi\)
−0.732906 + 0.680330i \(0.761836\pi\)
\(32\) 45.0000 0.248592
\(33\) −45.0000 −0.237379
\(34\) −252.000 −1.27111
\(35\) 0 0
\(36\) 9.00000 0.0416667
\(37\) −316.000 −1.40406 −0.702028 0.712149i \(-0.747722\pi\)
−0.702028 + 0.712149i \(0.747722\pi\)
\(38\) 48.0000 0.204911
\(39\) −192.000 −0.788323
\(40\) −63.0000 −0.249029
\(41\) 360.000 1.37128 0.685641 0.727940i \(-0.259522\pi\)
0.685641 + 0.727940i \(0.259522\pi\)
\(42\) 0 0
\(43\) 26.0000 0.0922084 0.0461042 0.998937i \(-0.485319\pi\)
0.0461042 + 0.998937i \(0.485319\pi\)
\(44\) −15.0000 −0.0513940
\(45\) −27.0000 −0.0894427
\(46\) 252.000 0.807725
\(47\) −30.0000 −0.0931053 −0.0465527 0.998916i \(-0.514824\pi\)
−0.0465527 + 0.998916i \(0.514824\pi\)
\(48\) −213.000 −0.640498
\(49\) 0 0
\(50\) 348.000 0.984293
\(51\) 252.000 0.691903
\(52\) −64.0000 −0.170677
\(53\) 363.000 0.940790 0.470395 0.882456i \(-0.344111\pi\)
0.470395 + 0.882456i \(0.344111\pi\)
\(54\) −81.0000 −0.204124
\(55\) 45.0000 0.110324
\(56\) 0 0
\(57\) −48.0000 −0.111540
\(58\) 891.000 2.01714
\(59\) −15.0000 −0.0330989 −0.0165494 0.999863i \(-0.505268\pi\)
−0.0165494 + 0.999863i \(0.505268\pi\)
\(60\) −9.00000 −0.0193649
\(61\) −118.000 −0.247678 −0.123839 0.992302i \(-0.539521\pi\)
−0.123839 + 0.992302i \(0.539521\pi\)
\(62\) 759.000 1.55473
\(63\) 0 0
\(64\) 433.000 0.845703
\(65\) 192.000 0.366380
\(66\) 135.000 0.251778
\(67\) −370.000 −0.674667 −0.337334 0.941385i \(-0.609525\pi\)
−0.337334 + 0.941385i \(0.609525\pi\)
\(68\) 84.0000 0.149801
\(69\) −252.000 −0.439670
\(70\) 0 0
\(71\) −342.000 −0.571661 −0.285831 0.958280i \(-0.592269\pi\)
−0.285831 + 0.958280i \(0.592269\pi\)
\(72\) 189.000 0.309359
\(73\) 362.000 0.580396 0.290198 0.956967i \(-0.406279\pi\)
0.290198 + 0.956967i \(0.406279\pi\)
\(74\) 948.000 1.48923
\(75\) −348.000 −0.535781
\(76\) −16.0000 −0.0241490
\(77\) 0 0
\(78\) 576.000 0.836143
\(79\) 467.000 0.665084 0.332542 0.943089i \(-0.392094\pi\)
0.332542 + 0.943089i \(0.392094\pi\)
\(80\) 213.000 0.297677
\(81\) 81.0000 0.111111
\(82\) −1080.00 −1.45446
\(83\) 477.000 0.630814 0.315407 0.948957i \(-0.397859\pi\)
0.315407 + 0.948957i \(0.397859\pi\)
\(84\) 0 0
\(85\) −252.000 −0.321568
\(86\) −78.0000 −0.0978018
\(87\) −891.000 −1.09799
\(88\) −315.000 −0.381581
\(89\) 906.000 1.07905 0.539527 0.841968i \(-0.318603\pi\)
0.539527 + 0.841968i \(0.318603\pi\)
\(90\) 81.0000 0.0948683
\(91\) 0 0
\(92\) −84.0000 −0.0951914
\(93\) −759.000 −0.846286
\(94\) 90.0000 0.0987531
\(95\) 48.0000 0.0518389
\(96\) 135.000 0.143525
\(97\) 503.000 0.526515 0.263257 0.964726i \(-0.415203\pi\)
0.263257 + 0.964726i \(0.415203\pi\)
\(98\) 0 0
\(99\) −135.000 −0.137051
\(100\) −116.000 −0.116000
\(101\) −1086.00 −1.06991 −0.534956 0.844880i \(-0.679672\pi\)
−0.534956 + 0.844880i \(0.679672\pi\)
\(102\) −756.000 −0.733874
\(103\) 1736.00 1.66071 0.830355 0.557235i \(-0.188138\pi\)
0.830355 + 0.557235i \(0.188138\pi\)
\(104\) −1344.00 −1.26721
\(105\) 0 0
\(106\) −1089.00 −0.997859
\(107\) −1353.00 −1.22242 −0.611212 0.791467i \(-0.709318\pi\)
−0.611212 + 0.791467i \(0.709318\pi\)
\(108\) 27.0000 0.0240563
\(109\) −370.000 −0.325134 −0.162567 0.986698i \(-0.551977\pi\)
−0.162567 + 0.986698i \(0.551977\pi\)
\(110\) −135.000 −0.117016
\(111\) −948.000 −0.810632
\(112\) 0 0
\(113\) −648.000 −0.539458 −0.269729 0.962936i \(-0.586934\pi\)
−0.269729 + 0.962936i \(0.586934\pi\)
\(114\) 144.000 0.118306
\(115\) 252.000 0.204340
\(116\) −297.000 −0.237722
\(117\) −576.000 −0.455139
\(118\) 45.0000 0.0351067
\(119\) 0 0
\(120\) −189.000 −0.143777
\(121\) −1106.00 −0.830954
\(122\) 354.000 0.262702
\(123\) 1080.00 0.791710
\(124\) −253.000 −0.183226
\(125\) 723.000 0.517337
\(126\) 0 0
\(127\) 377.000 0.263412 0.131706 0.991289i \(-0.457954\pi\)
0.131706 + 0.991289i \(0.457954\pi\)
\(128\) −1659.00 −1.14560
\(129\) 78.0000 0.0532366
\(130\) −576.000 −0.388604
\(131\) −651.000 −0.434184 −0.217092 0.976151i \(-0.569657\pi\)
−0.217092 + 0.976151i \(0.569657\pi\)
\(132\) −45.0000 −0.0296723
\(133\) 0 0
\(134\) 1110.00 0.715593
\(135\) −81.0000 −0.0516398
\(136\) 1764.00 1.11222
\(137\) −1770.00 −1.10381 −0.551903 0.833909i \(-0.686098\pi\)
−0.551903 + 0.833909i \(0.686098\pi\)
\(138\) 756.000 0.466341
\(139\) −1558.00 −0.950704 −0.475352 0.879796i \(-0.657679\pi\)
−0.475352 + 0.879796i \(0.657679\pi\)
\(140\) 0 0
\(141\) −90.0000 −0.0537544
\(142\) 1026.00 0.606338
\(143\) 960.000 0.561393
\(144\) −639.000 −0.369792
\(145\) 891.000 0.510300
\(146\) −1086.00 −0.615603
\(147\) 0 0
\(148\) −316.000 −0.175507
\(149\) 2454.00 1.34926 0.674629 0.738157i \(-0.264304\pi\)
0.674629 + 0.738157i \(0.264304\pi\)
\(150\) 1044.00 0.568282
\(151\) 1259.00 0.678516 0.339258 0.940693i \(-0.389824\pi\)
0.339258 + 0.940693i \(0.389824\pi\)
\(152\) −336.000 −0.179297
\(153\) 756.000 0.399470
\(154\) 0 0
\(155\) 759.000 0.393318
\(156\) −192.000 −0.0985404
\(157\) −196.000 −0.0996338 −0.0498169 0.998758i \(-0.515864\pi\)
−0.0498169 + 0.998758i \(0.515864\pi\)
\(158\) −1401.00 −0.705428
\(159\) 1089.00 0.543166
\(160\) −135.000 −0.0667043
\(161\) 0 0
\(162\) −243.000 −0.117851
\(163\) −1252.00 −0.601621 −0.300810 0.953684i \(-0.597257\pi\)
−0.300810 + 0.953684i \(0.597257\pi\)
\(164\) 360.000 0.171410
\(165\) 135.000 0.0636954
\(166\) −1431.00 −0.669079
\(167\) −2646.00 −1.22607 −0.613035 0.790056i \(-0.710051\pi\)
−0.613035 + 0.790056i \(0.710051\pi\)
\(168\) 0 0
\(169\) 1899.00 0.864360
\(170\) 756.000 0.341074
\(171\) −144.000 −0.0643974
\(172\) 26.0000 0.0115261
\(173\) −786.000 −0.345425 −0.172712 0.984972i \(-0.555253\pi\)
−0.172712 + 0.984972i \(0.555253\pi\)
\(174\) 2673.00 1.16460
\(175\) 0 0
\(176\) 1065.00 0.456122
\(177\) −45.0000 −0.0191096
\(178\) −2718.00 −1.14451
\(179\) 2892.00 1.20759 0.603794 0.797140i \(-0.293655\pi\)
0.603794 + 0.797140i \(0.293655\pi\)
\(180\) −27.0000 −0.0111803
\(181\) 1352.00 0.555212 0.277606 0.960695i \(-0.410459\pi\)
0.277606 + 0.960695i \(0.410459\pi\)
\(182\) 0 0
\(183\) −354.000 −0.142997
\(184\) −1764.00 −0.706760
\(185\) 948.000 0.376748
\(186\) 2277.00 0.897622
\(187\) −1260.00 −0.492729
\(188\) −30.0000 −0.0116382
\(189\) 0 0
\(190\) −144.000 −0.0549835
\(191\) 3912.00 1.48200 0.741001 0.671504i \(-0.234351\pi\)
0.741001 + 0.671504i \(0.234351\pi\)
\(192\) 1299.00 0.488267
\(193\) 1493.00 0.556832 0.278416 0.960461i \(-0.410191\pi\)
0.278416 + 0.960461i \(0.410191\pi\)
\(194\) −1509.00 −0.558453
\(195\) 576.000 0.211529
\(196\) 0 0
\(197\) −4086.00 −1.47774 −0.738872 0.673846i \(-0.764641\pi\)
−0.738872 + 0.673846i \(0.764641\pi\)
\(198\) 405.000 0.145364
\(199\) −3556.00 −1.26672 −0.633362 0.773855i \(-0.718326\pi\)
−0.633362 + 0.773855i \(0.718326\pi\)
\(200\) −2436.00 −0.861256
\(201\) −1110.00 −0.389519
\(202\) 3258.00 1.13481
\(203\) 0 0
\(204\) 252.000 0.0864879
\(205\) −1080.00 −0.367954
\(206\) −5208.00 −1.76145
\(207\) −756.000 −0.253844
\(208\) 4544.00 1.51476
\(209\) 240.000 0.0794313
\(210\) 0 0
\(211\) 1250.00 0.407837 0.203918 0.978988i \(-0.434632\pi\)
0.203918 + 0.978988i \(0.434632\pi\)
\(212\) 363.000 0.117599
\(213\) −1026.00 −0.330049
\(214\) 4059.00 1.29658
\(215\) −78.0000 −0.0247421
\(216\) 567.000 0.178609
\(217\) 0 0
\(218\) 1110.00 0.344856
\(219\) 1086.00 0.335092
\(220\) 45.0000 0.0137905
\(221\) −5376.00 −1.63633
\(222\) 2844.00 0.859805
\(223\) 425.000 0.127624 0.0638119 0.997962i \(-0.479674\pi\)
0.0638119 + 0.997962i \(0.479674\pi\)
\(224\) 0 0
\(225\) −1044.00 −0.309333
\(226\) 1944.00 0.572181
\(227\) 3855.00 1.12716 0.563580 0.826061i \(-0.309424\pi\)
0.563580 + 0.826061i \(0.309424\pi\)
\(228\) −48.0000 −0.0139424
\(229\) −2188.00 −0.631385 −0.315692 0.948862i \(-0.602237\pi\)
−0.315692 + 0.948862i \(0.602237\pi\)
\(230\) −756.000 −0.216735
\(231\) 0 0
\(232\) −6237.00 −1.76500
\(233\) 852.000 0.239555 0.119778 0.992801i \(-0.461782\pi\)
0.119778 + 0.992801i \(0.461782\pi\)
\(234\) 1728.00 0.482747
\(235\) 90.0000 0.0249828
\(236\) −15.0000 −0.00413736
\(237\) 1401.00 0.383986
\(238\) 0 0
\(239\) 5508.00 1.49072 0.745362 0.666660i \(-0.232277\pi\)
0.745362 + 0.666660i \(0.232277\pi\)
\(240\) 639.000 0.171864
\(241\) 791.000 0.211422 0.105711 0.994397i \(-0.466288\pi\)
0.105711 + 0.994397i \(0.466288\pi\)
\(242\) 3318.00 0.881360
\(243\) 243.000 0.0641500
\(244\) −118.000 −0.0309597
\(245\) 0 0
\(246\) −3240.00 −0.839735
\(247\) 1024.00 0.263788
\(248\) −5313.00 −1.36039
\(249\) 1431.00 0.364201
\(250\) −2169.00 −0.548718
\(251\) 5265.00 1.32400 0.662000 0.749504i \(-0.269708\pi\)
0.662000 + 0.749504i \(0.269708\pi\)
\(252\) 0 0
\(253\) 1260.00 0.313105
\(254\) −1131.00 −0.279391
\(255\) −756.000 −0.185657
\(256\) 1513.00 0.369385
\(257\) −6870.00 −1.66747 −0.833733 0.552168i \(-0.813801\pi\)
−0.833733 + 0.552168i \(0.813801\pi\)
\(258\) −234.000 −0.0564659
\(259\) 0 0
\(260\) 192.000 0.0457974
\(261\) −2673.00 −0.633925
\(262\) 1953.00 0.460522
\(263\) −222.000 −0.0520498 −0.0260249 0.999661i \(-0.508285\pi\)
−0.0260249 + 0.999661i \(0.508285\pi\)
\(264\) −945.000 −0.220306
\(265\) −1089.00 −0.252441
\(266\) 0 0
\(267\) 2718.00 0.622992
\(268\) −370.000 −0.0843334
\(269\) 7851.00 1.77949 0.889747 0.456454i \(-0.150881\pi\)
0.889747 + 0.456454i \(0.150881\pi\)
\(270\) 243.000 0.0547723
\(271\) 5183.00 1.16179 0.580895 0.813979i \(-0.302703\pi\)
0.580895 + 0.813979i \(0.302703\pi\)
\(272\) −5964.00 −1.32949
\(273\) 0 0
\(274\) 5310.00 1.17076
\(275\) 1740.00 0.381549
\(276\) −252.000 −0.0549588
\(277\) −4960.00 −1.07588 −0.537938 0.842985i \(-0.680796\pi\)
−0.537938 + 0.842985i \(0.680796\pi\)
\(278\) 4674.00 1.00837
\(279\) −2277.00 −0.488604
\(280\) 0 0
\(281\) −774.000 −0.164317 −0.0821583 0.996619i \(-0.526181\pi\)
−0.0821583 + 0.996619i \(0.526181\pi\)
\(282\) 270.000 0.0570151
\(283\) 3698.00 0.776761 0.388380 0.921499i \(-0.373035\pi\)
0.388380 + 0.921499i \(0.373035\pi\)
\(284\) −342.000 −0.0714576
\(285\) 144.000 0.0299292
\(286\) −2880.00 −0.595447
\(287\) 0 0
\(288\) 405.000 0.0828641
\(289\) 2143.00 0.436190
\(290\) −2673.00 −0.541255
\(291\) 1509.00 0.303983
\(292\) 362.000 0.0725495
\(293\) −6273.00 −1.25076 −0.625380 0.780321i \(-0.715056\pi\)
−0.625380 + 0.780321i \(0.715056\pi\)
\(294\) 0 0
\(295\) 45.0000 0.00888136
\(296\) −6636.00 −1.30307
\(297\) −405.000 −0.0791262
\(298\) −7362.00 −1.43110
\(299\) 5376.00 1.03981
\(300\) −348.000 −0.0669726
\(301\) 0 0
\(302\) −3777.00 −0.719675
\(303\) −3258.00 −0.617714
\(304\) 1136.00 0.214323
\(305\) 354.000 0.0664590
\(306\) −2268.00 −0.423702
\(307\) −1684.00 −0.313065 −0.156533 0.987673i \(-0.550032\pi\)
−0.156533 + 0.987673i \(0.550032\pi\)
\(308\) 0 0
\(309\) 5208.00 0.958812
\(310\) −2277.00 −0.417177
\(311\) −1320.00 −0.240676 −0.120338 0.992733i \(-0.538398\pi\)
−0.120338 + 0.992733i \(0.538398\pi\)
\(312\) −4032.00 −0.731625
\(313\) −8503.00 −1.53552 −0.767760 0.640737i \(-0.778629\pi\)
−0.767760 + 0.640737i \(0.778629\pi\)
\(314\) 588.000 0.105678
\(315\) 0 0
\(316\) 467.000 0.0831355
\(317\) −2577.00 −0.456589 −0.228295 0.973592i \(-0.573315\pi\)
−0.228295 + 0.973592i \(0.573315\pi\)
\(318\) −3267.00 −0.576114
\(319\) 4455.00 0.781919
\(320\) −1299.00 −0.226926
\(321\) −4059.00 −0.705767
\(322\) 0 0
\(323\) −1344.00 −0.231524
\(324\) 81.0000 0.0138889
\(325\) 7424.00 1.26711
\(326\) 3756.00 0.638115
\(327\) −1110.00 −0.187716
\(328\) 7560.00 1.27266
\(329\) 0 0
\(330\) −405.000 −0.0675591
\(331\) −484.000 −0.0803717 −0.0401859 0.999192i \(-0.512795\pi\)
−0.0401859 + 0.999192i \(0.512795\pi\)
\(332\) 477.000 0.0788517
\(333\) −2844.00 −0.468019
\(334\) 7938.00 1.30044
\(335\) 1110.00 0.181032
\(336\) 0 0
\(337\) −8359.00 −1.35117 −0.675584 0.737283i \(-0.736109\pi\)
−0.675584 + 0.737283i \(0.736109\pi\)
\(338\) −5697.00 −0.916793
\(339\) −1944.00 −0.311456
\(340\) −252.000 −0.0401959
\(341\) 3795.00 0.602671
\(342\) 432.000 0.0683038
\(343\) 0 0
\(344\) 546.000 0.0855766
\(345\) 756.000 0.117976
\(346\) 2358.00 0.366378
\(347\) −1860.00 −0.287752 −0.143876 0.989596i \(-0.545957\pi\)
−0.143876 + 0.989596i \(0.545957\pi\)
\(348\) −891.000 −0.137249
\(349\) −1918.00 −0.294178 −0.147089 0.989123i \(-0.546990\pi\)
−0.147089 + 0.989123i \(0.546990\pi\)
\(350\) 0 0
\(351\) −1728.00 −0.262774
\(352\) −675.000 −0.102209
\(353\) −3048.00 −0.459571 −0.229786 0.973241i \(-0.573803\pi\)
−0.229786 + 0.973241i \(0.573803\pi\)
\(354\) 135.000 0.0202688
\(355\) 1026.00 0.153393
\(356\) 906.000 0.134882
\(357\) 0 0
\(358\) −8676.00 −1.28084
\(359\) −30.0000 −0.00441042 −0.00220521 0.999998i \(-0.500702\pi\)
−0.00220521 + 0.999998i \(0.500702\pi\)
\(360\) −567.000 −0.0830098
\(361\) −6603.00 −0.962677
\(362\) −4056.00 −0.588891
\(363\) −3318.00 −0.479752
\(364\) 0 0
\(365\) −1086.00 −0.155737
\(366\) 1062.00 0.151671
\(367\) −11311.0 −1.60880 −0.804400 0.594088i \(-0.797513\pi\)
−0.804400 + 0.594088i \(0.797513\pi\)
\(368\) 5964.00 0.844823
\(369\) 3240.00 0.457094
\(370\) −2844.00 −0.399601
\(371\) 0 0
\(372\) −759.000 −0.105786
\(373\) 1208.00 0.167689 0.0838443 0.996479i \(-0.473280\pi\)
0.0838443 + 0.996479i \(0.473280\pi\)
\(374\) 3780.00 0.522618
\(375\) 2169.00 0.298684
\(376\) −630.000 −0.0864090
\(377\) 19008.0 2.59672
\(378\) 0 0
\(379\) 7640.00 1.03546 0.517731 0.855543i \(-0.326777\pi\)
0.517731 + 0.855543i \(0.326777\pi\)
\(380\) 48.0000 0.00647986
\(381\) 1131.00 0.152081
\(382\) −11736.0 −1.57190
\(383\) 12750.0 1.70103 0.850515 0.525951i \(-0.176290\pi\)
0.850515 + 0.525951i \(0.176290\pi\)
\(384\) −4977.00 −0.661410
\(385\) 0 0
\(386\) −4479.00 −0.590609
\(387\) 234.000 0.0307361
\(388\) 503.000 0.0658143
\(389\) 3126.00 0.407441 0.203720 0.979029i \(-0.434697\pi\)
0.203720 + 0.979029i \(0.434697\pi\)
\(390\) −1728.00 −0.224361
\(391\) −7056.00 −0.912627
\(392\) 0 0
\(393\) −1953.00 −0.250676
\(394\) 12258.0 1.56738
\(395\) −1401.00 −0.178461
\(396\) −135.000 −0.0171313
\(397\) −5932.00 −0.749921 −0.374960 0.927041i \(-0.622344\pi\)
−0.374960 + 0.927041i \(0.622344\pi\)
\(398\) 10668.0 1.34356
\(399\) 0 0
\(400\) 8236.00 1.02950
\(401\) 1608.00 0.200249 0.100124 0.994975i \(-0.468076\pi\)
0.100124 + 0.994975i \(0.468076\pi\)
\(402\) 3330.00 0.413148
\(403\) 16192.0 2.00144
\(404\) −1086.00 −0.133739
\(405\) −243.000 −0.0298142
\(406\) 0 0
\(407\) 4740.00 0.577280
\(408\) 5292.00 0.642140
\(409\) −4465.00 −0.539805 −0.269902 0.962888i \(-0.586991\pi\)
−0.269902 + 0.962888i \(0.586991\pi\)
\(410\) 3240.00 0.390274
\(411\) −5310.00 −0.637282
\(412\) 1736.00 0.207589
\(413\) 0 0
\(414\) 2268.00 0.269242
\(415\) −1431.00 −0.169265
\(416\) −2880.00 −0.339432
\(417\) −4674.00 −0.548889
\(418\) −720.000 −0.0842496
\(419\) −1584.00 −0.184686 −0.0923430 0.995727i \(-0.529436\pi\)
−0.0923430 + 0.995727i \(0.529436\pi\)
\(420\) 0 0
\(421\) −1330.00 −0.153967 −0.0769837 0.997032i \(-0.524529\pi\)
−0.0769837 + 0.997032i \(0.524529\pi\)
\(422\) −3750.00 −0.432576
\(423\) −270.000 −0.0310351
\(424\) 7623.00 0.873126
\(425\) −9744.00 −1.11213
\(426\) 3078.00 0.350069
\(427\) 0 0
\(428\) −1353.00 −0.152803
\(429\) 2880.00 0.324121
\(430\) 234.000 0.0262430
\(431\) 9588.00 1.07155 0.535775 0.844361i \(-0.320020\pi\)
0.535775 + 0.844361i \(0.320020\pi\)
\(432\) −1917.00 −0.213499
\(433\) 494.000 0.0548271 0.0274135 0.999624i \(-0.491273\pi\)
0.0274135 + 0.999624i \(0.491273\pi\)
\(434\) 0 0
\(435\) 2673.00 0.294622
\(436\) −370.000 −0.0406417
\(437\) 1344.00 0.147122
\(438\) −3258.00 −0.355418
\(439\) −16009.0 −1.74047 −0.870237 0.492634i \(-0.836034\pi\)
−0.870237 + 0.492634i \(0.836034\pi\)
\(440\) 945.000 0.102389
\(441\) 0 0
\(442\) 16128.0 1.73559
\(443\) 7773.00 0.833649 0.416824 0.908987i \(-0.363143\pi\)
0.416824 + 0.908987i \(0.363143\pi\)
\(444\) −948.000 −0.101329
\(445\) −2718.00 −0.289541
\(446\) −1275.00 −0.135365
\(447\) 7362.00 0.778995
\(448\) 0 0
\(449\) 864.000 0.0908122 0.0454061 0.998969i \(-0.485542\pi\)
0.0454061 + 0.998969i \(0.485542\pi\)
\(450\) 3132.00 0.328098
\(451\) −5400.00 −0.563805
\(452\) −648.000 −0.0674322
\(453\) 3777.00 0.391742
\(454\) −11565.0 −1.19553
\(455\) 0 0
\(456\) −1008.00 −0.103517
\(457\) 2519.00 0.257842 0.128921 0.991655i \(-0.458849\pi\)
0.128921 + 0.991655i \(0.458849\pi\)
\(458\) 6564.00 0.669685
\(459\) 2268.00 0.230634
\(460\) 252.000 0.0255425
\(461\) −342.000 −0.0345521 −0.0172761 0.999851i \(-0.505499\pi\)
−0.0172761 + 0.999851i \(0.505499\pi\)
\(462\) 0 0
\(463\) −4336.00 −0.435229 −0.217614 0.976035i \(-0.569828\pi\)
−0.217614 + 0.976035i \(0.569828\pi\)
\(464\) 21087.0 2.10978
\(465\) 2277.00 0.227082
\(466\) −2556.00 −0.254087
\(467\) 18636.0 1.84662 0.923310 0.384056i \(-0.125473\pi\)
0.923310 + 0.384056i \(0.125473\pi\)
\(468\) −576.000 −0.0568923
\(469\) 0 0
\(470\) −270.000 −0.0264982
\(471\) −588.000 −0.0575236
\(472\) −315.000 −0.0307183
\(473\) −390.000 −0.0379117
\(474\) −4203.00 −0.407279
\(475\) 1856.00 0.179282
\(476\) 0 0
\(477\) 3267.00 0.313597
\(478\) −16524.0 −1.58115
\(479\) 15078.0 1.43827 0.719135 0.694870i \(-0.244538\pi\)
0.719135 + 0.694870i \(0.244538\pi\)
\(480\) −405.000 −0.0385117
\(481\) 20224.0 1.91712
\(482\) −2373.00 −0.224247
\(483\) 0 0
\(484\) −1106.00 −0.103869
\(485\) −1509.00 −0.141279
\(486\) −729.000 −0.0680414
\(487\) 6221.00 0.578851 0.289425 0.957201i \(-0.406536\pi\)
0.289425 + 0.957201i \(0.406536\pi\)
\(488\) −2478.00 −0.229864
\(489\) −3756.00 −0.347346
\(490\) 0 0
\(491\) −7371.00 −0.677492 −0.338746 0.940878i \(-0.610003\pi\)
−0.338746 + 0.940878i \(0.610003\pi\)
\(492\) 1080.00 0.0989637
\(493\) −24948.0 −2.27911
\(494\) −3072.00 −0.279789
\(495\) 405.000 0.0367745
\(496\) 17963.0 1.62613
\(497\) 0 0
\(498\) −4293.00 −0.386293
\(499\) 4274.00 0.383428 0.191714 0.981451i \(-0.438595\pi\)
0.191714 + 0.981451i \(0.438595\pi\)
\(500\) 723.000 0.0646671
\(501\) −7938.00 −0.707872
\(502\) −15795.0 −1.40431
\(503\) −2520.00 −0.223382 −0.111691 0.993743i \(-0.535627\pi\)
−0.111691 + 0.993743i \(0.535627\pi\)
\(504\) 0 0
\(505\) 3258.00 0.287087
\(506\) −3780.00 −0.332098
\(507\) 5697.00 0.499039
\(508\) 377.000 0.0329265
\(509\) −14277.0 −1.24326 −0.621628 0.783313i \(-0.713528\pi\)
−0.621628 + 0.783313i \(0.713528\pi\)
\(510\) 2268.00 0.196919
\(511\) 0 0
\(512\) 8733.00 0.753804
\(513\) −432.000 −0.0371799
\(514\) 20610.0 1.76862
\(515\) −5208.00 −0.445615
\(516\) 78.0000 0.00665457
\(517\) 450.000 0.0382804
\(518\) 0 0
\(519\) −2358.00 −0.199431
\(520\) 4032.00 0.340029
\(521\) −6306.00 −0.530270 −0.265135 0.964211i \(-0.585417\pi\)
−0.265135 + 0.964211i \(0.585417\pi\)
\(522\) 8019.00 0.672379
\(523\) 8072.00 0.674883 0.337442 0.941346i \(-0.390438\pi\)
0.337442 + 0.941346i \(0.390438\pi\)
\(524\) −651.000 −0.0542730
\(525\) 0 0
\(526\) 666.000 0.0552072
\(527\) −21252.0 −1.75664
\(528\) 3195.00 0.263342
\(529\) −5111.00 −0.420071
\(530\) 3267.00 0.267754
\(531\) −135.000 −0.0110330
\(532\) 0 0
\(533\) −23040.0 −1.87237
\(534\) −8154.00 −0.660783
\(535\) 4059.00 0.328011
\(536\) −7770.00 −0.626143
\(537\) 8676.00 0.697201
\(538\) −23553.0 −1.88744
\(539\) 0 0
\(540\) −81.0000 −0.00645497
\(541\) −22858.0 −1.81653 −0.908264 0.418396i \(-0.862592\pi\)
−0.908264 + 0.418396i \(0.862592\pi\)
\(542\) −15549.0 −1.23226
\(543\) 4056.00 0.320552
\(544\) 3780.00 0.297916
\(545\) 1110.00 0.0872425
\(546\) 0 0
\(547\) −24724.0 −1.93258 −0.966291 0.257454i \(-0.917116\pi\)
−0.966291 + 0.257454i \(0.917116\pi\)
\(548\) −1770.00 −0.137976
\(549\) −1062.00 −0.0825593
\(550\) −5220.00 −0.404694
\(551\) 4752.00 0.367408
\(552\) −5292.00 −0.408048
\(553\) 0 0
\(554\) 14880.0 1.14114
\(555\) 2844.00 0.217515
\(556\) −1558.00 −0.118838
\(557\) −9843.00 −0.748764 −0.374382 0.927275i \(-0.622145\pi\)
−0.374382 + 0.927275i \(0.622145\pi\)
\(558\) 6831.00 0.518242
\(559\) −1664.00 −0.125903
\(560\) 0 0
\(561\) −3780.00 −0.284477
\(562\) 2322.00 0.174284
\(563\) −13371.0 −1.00092 −0.500462 0.865758i \(-0.666837\pi\)
−0.500462 + 0.865758i \(0.666837\pi\)
\(564\) −90.0000 −0.00671930
\(565\) 1944.00 0.144752
\(566\) −11094.0 −0.823879
\(567\) 0 0
\(568\) −7182.00 −0.530546
\(569\) −5232.00 −0.385478 −0.192739 0.981250i \(-0.561737\pi\)
−0.192739 + 0.981250i \(0.561737\pi\)
\(570\) −432.000 −0.0317447
\(571\) −14398.0 −1.05523 −0.527616 0.849483i \(-0.676914\pi\)
−0.527616 + 0.849483i \(0.676914\pi\)
\(572\) 960.000 0.0701742
\(573\) 11736.0 0.855634
\(574\) 0 0
\(575\) 9744.00 0.706701
\(576\) 3897.00 0.281901
\(577\) 19871.0 1.43369 0.716846 0.697231i \(-0.245585\pi\)
0.716846 + 0.697231i \(0.245585\pi\)
\(578\) −6429.00 −0.462649
\(579\) 4479.00 0.321487
\(580\) 891.000 0.0637875
\(581\) 0 0
\(582\) −4527.00 −0.322423
\(583\) −5445.00 −0.386808
\(584\) 7602.00 0.538652
\(585\) 1728.00 0.122127
\(586\) 18819.0 1.32663
\(587\) −16137.0 −1.13466 −0.567330 0.823491i \(-0.692024\pi\)
−0.567330 + 0.823491i \(0.692024\pi\)
\(588\) 0 0
\(589\) 4048.00 0.283183
\(590\) −135.000 −0.00942011
\(591\) −12258.0 −0.853176
\(592\) 22436.0 1.55762
\(593\) −21324.0 −1.47668 −0.738340 0.674428i \(-0.764390\pi\)
−0.738340 + 0.674428i \(0.764390\pi\)
\(594\) 1215.00 0.0839260
\(595\) 0 0
\(596\) 2454.00 0.168657
\(597\) −10668.0 −0.731344
\(598\) −16128.0 −1.10288
\(599\) −8646.00 −0.589760 −0.294880 0.955534i \(-0.595280\pi\)
−0.294880 + 0.955534i \(0.595280\pi\)
\(600\) −7308.00 −0.497246
\(601\) 11195.0 0.759823 0.379911 0.925023i \(-0.375954\pi\)
0.379911 + 0.925023i \(0.375954\pi\)
\(602\) 0 0
\(603\) −3330.00 −0.224889
\(604\) 1259.00 0.0848145
\(605\) 3318.00 0.222968
\(606\) 9774.00 0.655184
\(607\) −8971.00 −0.599871 −0.299935 0.953959i \(-0.596965\pi\)
−0.299935 + 0.953959i \(0.596965\pi\)
\(608\) −720.000 −0.0480261
\(609\) 0 0
\(610\) −1062.00 −0.0704904
\(611\) 1920.00 0.127127
\(612\) 756.000 0.0499338
\(613\) −12772.0 −0.841527 −0.420764 0.907170i \(-0.638238\pi\)
−0.420764 + 0.907170i \(0.638238\pi\)
\(614\) 5052.00 0.332056
\(615\) −3240.00 −0.212438
\(616\) 0 0
\(617\) 12762.0 0.832705 0.416352 0.909203i \(-0.363308\pi\)
0.416352 + 0.909203i \(0.363308\pi\)
\(618\) −15624.0 −1.01697
\(619\) 12842.0 0.833867 0.416933 0.908937i \(-0.363105\pi\)
0.416933 + 0.908937i \(0.363105\pi\)
\(620\) 759.000 0.0491648
\(621\) −2268.00 −0.146557
\(622\) 3960.00 0.255276
\(623\) 0 0
\(624\) 13632.0 0.874546
\(625\) 12331.0 0.789184
\(626\) 25509.0 1.62867
\(627\) 720.000 0.0458597
\(628\) −196.000 −0.0124542
\(629\) −26544.0 −1.68264
\(630\) 0 0
\(631\) 21365.0 1.34790 0.673952 0.738775i \(-0.264596\pi\)
0.673952 + 0.738775i \(0.264596\pi\)
\(632\) 9807.00 0.617249
\(633\) 3750.00 0.235465
\(634\) 7731.00 0.484286
\(635\) −1131.00 −0.0706809
\(636\) 1089.00 0.0678957
\(637\) 0 0
\(638\) −13365.0 −0.829350
\(639\) −3078.00 −0.190554
\(640\) 4977.00 0.307396
\(641\) 8274.00 0.509834 0.254917 0.966963i \(-0.417952\pi\)
0.254917 + 0.966963i \(0.417952\pi\)
\(642\) 12177.0 0.748579
\(643\) 27998.0 1.71716 0.858580 0.512680i \(-0.171347\pi\)
0.858580 + 0.512680i \(0.171347\pi\)
\(644\) 0 0
\(645\) −234.000 −0.0142849
\(646\) 4032.00 0.245568
\(647\) −17466.0 −1.06130 −0.530649 0.847592i \(-0.678052\pi\)
−0.530649 + 0.847592i \(0.678052\pi\)
\(648\) 1701.00 0.103120
\(649\) 225.000 0.0136087
\(650\) −22272.0 −1.34397
\(651\) 0 0
\(652\) −1252.00 −0.0752026
\(653\) 2157.00 0.129265 0.0646324 0.997909i \(-0.479413\pi\)
0.0646324 + 0.997909i \(0.479413\pi\)
\(654\) 3330.00 0.199103
\(655\) 1953.00 0.116504
\(656\) −25560.0 −1.52127
\(657\) 3258.00 0.193465
\(658\) 0 0
\(659\) 19944.0 1.17892 0.589460 0.807798i \(-0.299341\pi\)
0.589460 + 0.807798i \(0.299341\pi\)
\(660\) 135.000 0.00796192
\(661\) 27506.0 1.61855 0.809273 0.587432i \(-0.199861\pi\)
0.809273 + 0.587432i \(0.199861\pi\)
\(662\) 1452.00 0.0852471
\(663\) −16128.0 −0.944735
\(664\) 10017.0 0.585444
\(665\) 0 0
\(666\) 8532.00 0.496409
\(667\) 24948.0 1.44826
\(668\) −2646.00 −0.153259
\(669\) 1275.00 0.0736836
\(670\) −3330.00 −0.192014
\(671\) 1770.00 0.101833
\(672\) 0 0
\(673\) −19123.0 −1.09530 −0.547650 0.836707i \(-0.684478\pi\)
−0.547650 + 0.836707i \(0.684478\pi\)
\(674\) 25077.0 1.43313
\(675\) −3132.00 −0.178594
\(676\) 1899.00 0.108045
\(677\) 13857.0 0.786658 0.393329 0.919398i \(-0.371323\pi\)
0.393329 + 0.919398i \(0.371323\pi\)
\(678\) 5832.00 0.330349
\(679\) 0 0
\(680\) −5292.00 −0.298440
\(681\) 11565.0 0.650766
\(682\) −11385.0 −0.639229
\(683\) −22245.0 −1.24624 −0.623120 0.782127i \(-0.714135\pi\)
−0.623120 + 0.782127i \(0.714135\pi\)
\(684\) −144.000 −0.00804967
\(685\) 5310.00 0.296182
\(686\) 0 0
\(687\) −6564.00 −0.364530
\(688\) −1846.00 −0.102294
\(689\) −23232.0 −1.28457
\(690\) −2268.00 −0.125132
\(691\) −640.000 −0.0352341 −0.0176170 0.999845i \(-0.505608\pi\)
−0.0176170 + 0.999845i \(0.505608\pi\)
\(692\) −786.000 −0.0431781
\(693\) 0 0
\(694\) 5580.00 0.305207
\(695\) 4674.00 0.255101
\(696\) −18711.0 −1.01902
\(697\) 30240.0 1.64336
\(698\) 5754.00 0.312023
\(699\) 2556.00 0.138307
\(700\) 0 0
\(701\) −15561.0 −0.838418 −0.419209 0.907890i \(-0.637693\pi\)
−0.419209 + 0.907890i \(0.637693\pi\)
\(702\) 5184.00 0.278714
\(703\) 5056.00 0.271253
\(704\) −6495.00 −0.347712
\(705\) 270.000 0.0144238
\(706\) 9144.00 0.487449
\(707\) 0 0
\(708\) −45.0000 −0.00238871
\(709\) 5534.00 0.293136 0.146568 0.989201i \(-0.453177\pi\)
0.146568 + 0.989201i \(0.453177\pi\)
\(710\) −3078.00 −0.162698
\(711\) 4203.00 0.221695
\(712\) 19026.0 1.00145
\(713\) 21252.0 1.11626
\(714\) 0 0
\(715\) −2880.00 −0.150638
\(716\) 2892.00 0.150948
\(717\) 16524.0 0.860670
\(718\) 90.0000 0.00467795
\(719\) −21846.0 −1.13313 −0.566564 0.824018i \(-0.691727\pi\)
−0.566564 + 0.824018i \(0.691727\pi\)
\(720\) 1917.00 0.0992255
\(721\) 0 0
\(722\) 19809.0 1.02107
\(723\) 2373.00 0.122065
\(724\) 1352.00 0.0694015
\(725\) 34452.0 1.76485
\(726\) 9954.00 0.508853
\(727\) −11089.0 −0.565706 −0.282853 0.959163i \(-0.591281\pi\)
−0.282853 + 0.959163i \(0.591281\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 3258.00 0.165184
\(731\) 2184.00 0.110504
\(732\) −354.000 −0.0178746
\(733\) 11762.0 0.592687 0.296343 0.955081i \(-0.404233\pi\)
0.296343 + 0.955081i \(0.404233\pi\)
\(734\) 33933.0 1.70639
\(735\) 0 0
\(736\) −3780.00 −0.189311
\(737\) 5550.00 0.277391
\(738\) −9720.00 −0.484821
\(739\) −22726.0 −1.13124 −0.565622 0.824665i \(-0.691364\pi\)
−0.565622 + 0.824665i \(0.691364\pi\)
\(740\) 948.000 0.0470935
\(741\) 3072.00 0.152298
\(742\) 0 0
\(743\) 6678.00 0.329734 0.164867 0.986316i \(-0.447281\pi\)
0.164867 + 0.986316i \(0.447281\pi\)
\(744\) −15939.0 −0.785419
\(745\) −7362.00 −0.362044
\(746\) −3624.00 −0.177861
\(747\) 4293.00 0.210271
\(748\) −1260.00 −0.0615911
\(749\) 0 0
\(750\) −6507.00 −0.316803
\(751\) −19987.0 −0.971153 −0.485577 0.874194i \(-0.661390\pi\)
−0.485577 + 0.874194i \(0.661390\pi\)
\(752\) 2130.00 0.103289
\(753\) 15795.0 0.764411
\(754\) −57024.0 −2.75423
\(755\) −3777.00 −0.182065
\(756\) 0 0
\(757\) 314.000 0.0150760 0.00753799 0.999972i \(-0.497601\pi\)
0.00753799 + 0.999972i \(0.497601\pi\)
\(758\) −22920.0 −1.09827
\(759\) 3780.00 0.180771
\(760\) 1008.00 0.0481105
\(761\) −11496.0 −0.547608 −0.273804 0.961786i \(-0.588282\pi\)
−0.273804 + 0.961786i \(0.588282\pi\)
\(762\) −3393.00 −0.161306
\(763\) 0 0
\(764\) 3912.00 0.185250
\(765\) −2268.00 −0.107189
\(766\) −38250.0 −1.80421
\(767\) 960.000 0.0451937
\(768\) 4539.00 0.213264
\(769\) 2765.00 0.129660 0.0648299 0.997896i \(-0.479350\pi\)
0.0648299 + 0.997896i \(0.479350\pi\)
\(770\) 0 0
\(771\) −20610.0 −0.962712
\(772\) 1493.00 0.0696039
\(773\) −14046.0 −0.653557 −0.326778 0.945101i \(-0.605963\pi\)
−0.326778 + 0.945101i \(0.605963\pi\)
\(774\) −702.000 −0.0326006
\(775\) 29348.0 1.36027
\(776\) 10563.0 0.488646
\(777\) 0 0
\(778\) −9378.00 −0.432156
\(779\) −5760.00 −0.264921
\(780\) 576.000 0.0264412
\(781\) 5130.00 0.235039
\(782\) 21168.0 0.967987
\(783\) −8019.00 −0.365997
\(784\) 0 0
\(785\) 588.000 0.0267345
\(786\) 5859.00 0.265882
\(787\) −18514.0 −0.838568 −0.419284 0.907855i \(-0.637719\pi\)
−0.419284 + 0.907855i \(0.637719\pi\)
\(788\) −4086.00 −0.184718
\(789\) −666.000 −0.0300510
\(790\) 4203.00 0.189286
\(791\) 0 0
\(792\) −2835.00 −0.127194
\(793\) 7552.00 0.338183
\(794\) 17796.0 0.795411
\(795\) −3267.00 −0.145747
\(796\) −3556.00 −0.158341
\(797\) −27495.0 −1.22199 −0.610993 0.791636i \(-0.709230\pi\)
−0.610993 + 0.791636i \(0.709230\pi\)
\(798\) 0 0
\(799\) −2520.00 −0.111578
\(800\) −5220.00 −0.230694
\(801\) 8154.00 0.359685
\(802\) −4824.00 −0.212396
\(803\) −5430.00 −0.238631
\(804\) −1110.00 −0.0486899
\(805\) 0 0
\(806\) −48576.0 −2.12285
\(807\) 23553.0 1.02739
\(808\) −22806.0 −0.992961
\(809\) −7944.00 −0.345236 −0.172618 0.984989i \(-0.555223\pi\)
−0.172618 + 0.984989i \(0.555223\pi\)
\(810\) 729.000 0.0316228
\(811\) −28942.0 −1.25313 −0.626567 0.779368i \(-0.715540\pi\)
−0.626567 + 0.779368i \(0.715540\pi\)
\(812\) 0 0
\(813\) 15549.0 0.670759
\(814\) −14220.0 −0.612298
\(815\) 3756.00 0.161432
\(816\) −17892.0 −0.767580
\(817\) −416.000 −0.0178140
\(818\) 13395.0 0.572549
\(819\) 0 0
\(820\) −1080.00 −0.0459942
\(821\) 8187.00 0.348025 0.174012 0.984743i \(-0.444327\pi\)
0.174012 + 0.984743i \(0.444327\pi\)
\(822\) 15930.0 0.675940
\(823\) −280.000 −0.0118593 −0.00592964 0.999982i \(-0.501887\pi\)
−0.00592964 + 0.999982i \(0.501887\pi\)
\(824\) 36456.0 1.54127
\(825\) 5220.00 0.220287
\(826\) 0 0
\(827\) 25317.0 1.06452 0.532260 0.846581i \(-0.321343\pi\)
0.532260 + 0.846581i \(0.321343\pi\)
\(828\) −756.000 −0.0317305
\(829\) 15320.0 0.641840 0.320920 0.947106i \(-0.396008\pi\)
0.320920 + 0.947106i \(0.396008\pi\)
\(830\) 4293.00 0.179533
\(831\) −14880.0 −0.621157
\(832\) −27712.0 −1.15474
\(833\) 0 0
\(834\) 14022.0 0.582185
\(835\) 7938.00 0.328989
\(836\) 240.000 0.00992892
\(837\) −6831.00 −0.282095
\(838\) 4752.00 0.195889
\(839\) 34092.0 1.40284 0.701422 0.712746i \(-0.252549\pi\)
0.701422 + 0.712746i \(0.252549\pi\)
\(840\) 0 0
\(841\) 63820.0 2.61675
\(842\) 3990.00 0.163307
\(843\) −2322.00 −0.0948682
\(844\) 1250.00 0.0509796
\(845\) −5697.00 −0.231932
\(846\) 810.000 0.0329177
\(847\) 0 0
\(848\) −25773.0 −1.04369
\(849\) 11094.0 0.448463
\(850\) 29232.0 1.17959
\(851\) 26544.0 1.06923
\(852\) −1026.00 −0.0412561
\(853\) −7378.00 −0.296152 −0.148076 0.988976i \(-0.547308\pi\)
−0.148076 + 0.988976i \(0.547308\pi\)
\(854\) 0 0
\(855\) 432.000 0.0172796
\(856\) −28413.0 −1.13451
\(857\) −15594.0 −0.621565 −0.310782 0.950481i \(-0.600591\pi\)
−0.310782 + 0.950481i \(0.600591\pi\)
\(858\) −8640.00 −0.343782
\(859\) −30538.0 −1.21297 −0.606486 0.795094i \(-0.707421\pi\)
−0.606486 + 0.795094i \(0.707421\pi\)
\(860\) −78.0000 −0.00309277
\(861\) 0 0
\(862\) −28764.0 −1.13655
\(863\) −822.000 −0.0324232 −0.0162116 0.999869i \(-0.505161\pi\)
−0.0162116 + 0.999869i \(0.505161\pi\)
\(864\) 1215.00 0.0478416
\(865\) 2358.00 0.0926872
\(866\) −1482.00 −0.0581529
\(867\) 6429.00 0.251834
\(868\) 0 0
\(869\) −7005.00 −0.273450
\(870\) −8019.00 −0.312494
\(871\) 23680.0 0.921201
\(872\) −7770.00 −0.301749
\(873\) 4527.00 0.175505
\(874\) −4032.00 −0.156046
\(875\) 0 0
\(876\) 1086.00 0.0418865
\(877\) −41824.0 −1.61037 −0.805186 0.593022i \(-0.797935\pi\)
−0.805186 + 0.593022i \(0.797935\pi\)
\(878\) 48027.0 1.84605
\(879\) −18819.0 −0.722126
\(880\) −3195.00 −0.122390
\(881\) −46098.0 −1.76286 −0.881431 0.472313i \(-0.843419\pi\)
−0.881431 + 0.472313i \(0.843419\pi\)
\(882\) 0 0
\(883\) 21008.0 0.800652 0.400326 0.916373i \(-0.368897\pi\)
0.400326 + 0.916373i \(0.368897\pi\)
\(884\) −5376.00 −0.204541
\(885\) 135.000 0.00512766
\(886\) −23319.0 −0.884218
\(887\) 24036.0 0.909865 0.454932 0.890526i \(-0.349663\pi\)
0.454932 + 0.890526i \(0.349663\pi\)
\(888\) −19908.0 −0.752330
\(889\) 0 0
\(890\) 8154.00 0.307104
\(891\) −1215.00 −0.0456835
\(892\) 425.000 0.0159530
\(893\) 480.000 0.0179872
\(894\) −22086.0 −0.826249
\(895\) −8676.00 −0.324030
\(896\) 0 0
\(897\) 16128.0 0.600332
\(898\) −2592.00 −0.0963209
\(899\) 75141.0 2.78764
\(900\) −1044.00 −0.0386667
\(901\) 30492.0 1.12745
\(902\) 16200.0 0.598006
\(903\) 0 0
\(904\) −13608.0 −0.500659
\(905\) −4056.00 −0.148979
\(906\) −11331.0 −0.415505
\(907\) 13292.0 0.486608 0.243304 0.969950i \(-0.421769\pi\)
0.243304 + 0.969950i \(0.421769\pi\)
\(908\) 3855.00 0.140895
\(909\) −9774.00 −0.356637
\(910\) 0 0
\(911\) −9306.00 −0.338443 −0.169221 0.985578i \(-0.554125\pi\)
−0.169221 + 0.985578i \(0.554125\pi\)
\(912\) 3408.00 0.123739
\(913\) −7155.00 −0.259360
\(914\) −7557.00 −0.273483
\(915\) 1062.00 0.0383701
\(916\) −2188.00 −0.0789231
\(917\) 0 0
\(918\) −6804.00 −0.244625
\(919\) 16496.0 0.592114 0.296057 0.955170i \(-0.404328\pi\)
0.296057 + 0.955170i \(0.404328\pi\)
\(920\) 5292.00 0.189644
\(921\) −5052.00 −0.180748
\(922\) 1026.00 0.0366481
\(923\) 21888.0 0.780555
\(924\) 0 0
\(925\) 36656.0 1.30296
\(926\) 13008.0 0.461630
\(927\) 15624.0 0.553570
\(928\) −13365.0 −0.472767
\(929\) 14154.0 0.499868 0.249934 0.968263i \(-0.419591\pi\)
0.249934 + 0.968263i \(0.419591\pi\)
\(930\) −6831.00 −0.240857
\(931\) 0 0
\(932\) 852.000 0.0299444
\(933\) −3960.00 −0.138955
\(934\) −55908.0 −1.95864
\(935\) 3780.00 0.132213
\(936\) −12096.0 −0.422404
\(937\) −3781.00 −0.131825 −0.0659124 0.997825i \(-0.520996\pi\)
−0.0659124 + 0.997825i \(0.520996\pi\)
\(938\) 0 0
\(939\) −25509.0 −0.886533
\(940\) 90.0000 0.00312285
\(941\) −25863.0 −0.895972 −0.447986 0.894041i \(-0.647859\pi\)
−0.447986 + 0.894041i \(0.647859\pi\)
\(942\) 1764.00 0.0610130
\(943\) −30240.0 −1.04427
\(944\) 1065.00 0.0367191
\(945\) 0 0
\(946\) 1170.00 0.0402114
\(947\) −42384.0 −1.45438 −0.727188 0.686438i \(-0.759173\pi\)
−0.727188 + 0.686438i \(0.759173\pi\)
\(948\) 1401.00 0.0479983
\(949\) −23168.0 −0.792482
\(950\) −5568.00 −0.190158
\(951\) −7731.00 −0.263612
\(952\) 0 0
\(953\) 10530.0 0.357923 0.178961 0.983856i \(-0.442726\pi\)
0.178961 + 0.983856i \(0.442726\pi\)
\(954\) −9801.00 −0.332620
\(955\) −11736.0 −0.397663
\(956\) 5508.00 0.186340
\(957\) 13365.0 0.451441
\(958\) −45234.0 −1.52552
\(959\) 0 0
\(960\) −3897.00 −0.131016
\(961\) 34218.0 1.14860
\(962\) −60672.0 −2.03341
\(963\) −12177.0 −0.407475
\(964\) 791.000 0.0264278
\(965\) −4479.00 −0.149414
\(966\) 0 0
\(967\) −38341.0 −1.27504 −0.637520 0.770434i \(-0.720040\pi\)
−0.637520 + 0.770434i \(0.720040\pi\)
\(968\) −23226.0 −0.771190
\(969\) −4032.00 −0.133670
\(970\) 4527.00 0.149849
\(971\) 1923.00 0.0635551 0.0317776 0.999495i \(-0.489883\pi\)
0.0317776 + 0.999495i \(0.489883\pi\)
\(972\) 243.000 0.00801875
\(973\) 0 0
\(974\) −18663.0 −0.613964
\(975\) 22272.0 0.731564
\(976\) 8378.00 0.274768
\(977\) 57090.0 1.86947 0.934734 0.355347i \(-0.115637\pi\)
0.934734 + 0.355347i \(0.115637\pi\)
\(978\) 11268.0 0.368416
\(979\) −13590.0 −0.443655
\(980\) 0 0
\(981\) −3330.00 −0.108378
\(982\) 22113.0 0.718589
\(983\) 5484.00 0.177937 0.0889687 0.996034i \(-0.471643\pi\)
0.0889687 + 0.996034i \(0.471643\pi\)
\(984\) 22680.0 0.734768
\(985\) 12258.0 0.396520
\(986\) 74844.0 2.41736
\(987\) 0 0
\(988\) 1024.00 0.0329735
\(989\) −2184.00 −0.0702196
\(990\) −1215.00 −0.0390053
\(991\) −22465.0 −0.720105 −0.360053 0.932932i \(-0.617241\pi\)
−0.360053 + 0.932932i \(0.617241\pi\)
\(992\) −11385.0 −0.364389
\(993\) −1452.00 −0.0464026
\(994\) 0 0
\(995\) 10668.0 0.339898
\(996\) 1431.00 0.0455251
\(997\) 29366.0 0.932829 0.466415 0.884566i \(-0.345546\pi\)
0.466415 + 0.884566i \(0.345546\pi\)
\(998\) −12822.0 −0.406687
\(999\) −8532.00 −0.270211
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 147.4.a.b.1.1 1
3.2 odd 2 441.4.a.l.1.1 1
4.3 odd 2 2352.4.a.i.1.1 1
7.2 even 3 21.4.e.a.4.1 2
7.3 odd 6 147.4.e.h.79.1 2
7.4 even 3 21.4.e.a.16.1 yes 2
7.5 odd 6 147.4.e.h.67.1 2
7.6 odd 2 147.4.a.a.1.1 1
21.2 odd 6 63.4.e.a.46.1 2
21.5 even 6 441.4.e.c.361.1 2
21.11 odd 6 63.4.e.a.37.1 2
21.17 even 6 441.4.e.c.226.1 2
21.20 even 2 441.4.a.k.1.1 1
28.11 odd 6 336.4.q.e.289.1 2
28.23 odd 6 336.4.q.e.193.1 2
28.27 even 2 2352.4.a.bd.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
21.4.e.a.4.1 2 7.2 even 3
21.4.e.a.16.1 yes 2 7.4 even 3
63.4.e.a.37.1 2 21.11 odd 6
63.4.e.a.46.1 2 21.2 odd 6
147.4.a.a.1.1 1 7.6 odd 2
147.4.a.b.1.1 1 1.1 even 1 trivial
147.4.e.h.67.1 2 7.5 odd 6
147.4.e.h.79.1 2 7.3 odd 6
336.4.q.e.193.1 2 28.23 odd 6
336.4.q.e.289.1 2 28.11 odd 6
441.4.a.k.1.1 1 21.20 even 2
441.4.a.l.1.1 1 3.2 odd 2
441.4.e.c.226.1 2 21.17 even 6
441.4.e.c.361.1 2 21.5 even 6
2352.4.a.i.1.1 1 4.3 odd 2
2352.4.a.bd.1.1 1 28.27 even 2